Optimal Statistical and Computational Rates for One Bit Matrix Completion

We consider one bit matrix completion under rank constraint. We present an estimator based on rank constrained maximum likelihood estimation, and an e cient greedy algorithm to solve it approximately based on an extension of conditional gradient descent. The output of the proposed algorithm converges at a linear rate to the underlying true low-rank matrix up to the optimal statistical estimation error rate, i.e., O( p rn log(n)/|⌦|), where n is the dimension of the underlying matrix and |⌦| is the number of observed entries. Our work establishes the first computationally efficient approach with provable guarantee for optimal estimation in one bit matrix completion. Our theory is supported by thorough numerical results.